The Franke Filtration of the Spaces of Automorphic Forms on the Symplectic Group of Rank Two

The Franke filtration is a finite filtration of the space of automorphic forms on a connected reductive linear algebraic group defined over an algebraic number field. The main feature of the filtration is that its quotients can be described in terms of parabolically induced representations using the main values of derivatives of Eisenstein series and the residues of these. The goal of this paper is to provide a complete explicit description of the Franke filtration of the space of automorphic forms on the symplectic group of rank two. The approach is different from the original approach of Franke, and takes into account the full cuspidal support of automorphic forms, that is, the cuspidal automorphic representation from which the Eisenstein series is built and the evaluation point at which it is evaluated. This considerably simplifies the exposition and makes it possible to obtain very explicit results and reveal the phenomena present in the filtration. The considered low rank case exhibits many of the properties and phenomena present in the cases of arbitrary rank. The explicit description of the Franke filtration in this case has important implications and applications in cohomology of congruence subgroups related to the Hilbert–Siegel modular forms of degree two, the Hilbert–Siegel modular variety of degree two and the corresponding Shimura variety.